# Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry

arXiv: https://arxiv.org/abs/2110.10464 · HTML: https://mlsanity.com/paper/2110.10464/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 20 Oct 2021.

## Builds on (most cited first, 17 of 17)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Fast Optimal Transport Averaging of Neuroimaging Data](https://mlsanity.com/paper/1503.08596/) (2015)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization](https://mlsanity.com/paper/1706.03267/) (2017)
- [Riemannian preconditioning](https://mlsanity.com/paper/1405.6055/) (2014)
- [Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation](https://mlsanity.com/paper/1306.3212/) (2013)
- [Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognition](https://mlsanity.com/paper/1608.04914/) (2016)
- [A Unified Framework for Domain Adaptation using Metric Learning on Manifolds](https://mlsanity.com/paper/1804.10834/) (2018)
- [Projection Robust Wasserstein Barycenters](https://mlsanity.com/paper/2102.03390/) (2021)
- [Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences](https://mlsanity.com/paper/1409.0083/) (2014)

## Built on by (most cited first, 10 of 10)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)
- [Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry](https://mlsanity.com/paper/2407.02607/) (2024)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning](https://mlsanity.com/paper/2605.04406/) (2026)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (2026)
- [Towards Robust EEG Decoding Based on Riemannian Self-Attention](https://mlsanity.com/paper/2606.25456/) (2026)
- [BrainLinear: A Linear Model for Brain Network Analysis in Sparse Tangent Subspaces](https://mlsanity.com/paper/2608.15266/) (2026)
- [Optimization over covariance matrices with a parameterized metric](https://mlsanity.com/paper/2609.17089/) (2026)

## Abstract (arXiv)

Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix M such that when M = I, we recover the BW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry which makes it amenable to various machine learning applications. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.